For the basil mayo: In a food processor, combine the oil, garlic and basil and process until the basil is finely chopped. Add the mayonnaise and process until smooth. Refrigerate.
For the cornmeal breading: In a bowl, mix the cornmeal, flour, Cajun spice and salt until combined.
For the fried green tomatoes: Heat the vegetable oil in a cast-iron skillet over medium-low heat until it reaches a temperature of 350 degrees F. Slice the green tomatoes into 12 to 16 slices, each about 1/4-inch thick.
To bread the tomato slices, place the flour in one bowl, the buttermilk in another and the cornmeal breading in another. In batches, place the tomatoes in the flour and shake off the excess. Then place them in the buttermilk and coat them completely. Finally, place them in the cornmeal breading and cover them completely in the cornmeal.
Place 4 to 6 of the tomato slices in the hot oil and fry them until golden brown, about 2 minutes per side. Remove the slices and drain them on paper towels. Continue frying until all the tomatoes are done.
To assemble the sandwiches, spread some of the basil mayo on each slice of bread. Divide the arugula, cheese, onions, fried green tomatoes and some of the tomato jam among 4 of the bread slices. Top with the remaining slices and serve.